This paper contains a short and simplified proof of desingularization over fields of characteristic zero, together with various applications to other problems in algebraic geometry (among others, the study of the behavior of desingularization of families of embedded schemes, and a formulation of desingularization which is stronger than Hironaka's). Our proof avoids the use of the Hilbert-Samuel function and Hironaka's notion of normal flatness: First we define a procedure for principalization of ideals (i.e. a procedure to make an ideal invertible), and then we show that desingularization of a closed subscheme $X$ is achieved by using the procedure of principalization for the ideal ${\mathcal I}(X)$ associated to the embedded scheme $X$. The paper intends to be an introduction to the subject, focused on the motivation of ideas used in this new approach, and particularly on applications, some of which do not follow from Hironaka's proof.

We obtain a maximum principle at infinity for solutions of a class of nonlinear singular elliptic differential inequalities on Riemannian manifolds under the sole geometrical assumptions of volume growth conditions. In the case of the Laplace-Beltrami operator we relate our results to stochastic completeness and parabolicity of the manifold.

We give several new characterizations of the dual of the dyadic Hardy space $H^{1,d}(\mathbb{T}^2)$, the so-called dyadic BMO space in two variables and denoted ${\mathrm{BMO}}^{\mathit d}_{prod}}$. These include characterizations in terms of Haar multipliers, in terms of the ``symmetrised paraproduct'' $\Lambda_b$, in terms of the rectangular BMO norms of the iterated ``sweeps'', and in terms of nested commutators with dyadic martingale transforms. We further explore the connection between ${\mathrm{BMO}}^{\mathit d}_{prod}}$ and John-Nirenberg type inequalities, and study a scale of rectangular BMO spaces.

Let us consider the class of ``nonvariational uniformly hypoelliptic operators'': $$ Lu\equiv\sum_{i,j=1}^{q}a_{ij} (x) X_{i} X_{j} u $$ where: $X_1,X_2,\ldots,X_q$ is a system of H\"ormander vector fields in $\mathbb{R}^{n}$ ($n>q$), $\{a_{ij}\}$ is a $q\times q$ uniformly elliptic matrix, and the functions $a_{ij} (x)$ are continuous, with a suitable control on the modulus of continuity. We prove that: $$ \| X_{i} X_{j} u \|_{BMO(\Omega^{\prime})} \leq c \left\{\left\|Lu\right\|_{BMO(\Omega)} + \left\| u\right\|_{BMO(\Omega)} \right\} $$ for domains $\Omega^{\prime}\subset\subset\Omega$ that are regular in a suitable sense. Moreover, the space $BMO$ in the above estimate can be replaced with a scale of spaces of the kind studied by Spanne. To get this estimate, several results are proved, regarding singular and fractional integrals on general spaces of homogeneous type, in relation with function spaces of $BMO$ type.

We consider quasilinear operators $T$ of {\it joint weak type} $(a,b;p,q)$ (in the sense of [Bennett, Sharpley: Interpolation of operators, Academic Press, 1988]) and study their properties on spaces $L_{\varphi,E}$ with the norm $\|\varphi(t)f^*(t) \|_{\tilde E}$, where $\tilde E$ is arbitrary rearrangement-invariant space with respect to the measure $dt/t$. A space $L_{\varphi,E}$ is said to be ``close" to one of the endpoints of interpolation if the corresponding Boyd index of this space is equal to $1/a$ or to $1/p$. For all possible kinds of such ``closeness", we give sharp estimates for the function $\psi(t)$ so as to obtain that every $T:L_{\varphi,E}\to L_{\psi,E}$.

Suppose that, for each point $x$ in a given subset $E \subset \mathbb{R}^n$, we are given an $m$-jet $f(x)$ and a convex, symmetric set $\sigma(x)$ of $m$-jets at $x$. We ask whether there exist a function $F \in C^{m , \omega} ( \mathbb{R}^n )$ and a finite constant $M$, such that the $m$-jet of $F$ at $x$ belongs to $f ( x ) + M \sigma ( x )$ for all $x \in E$. We give a necessary and sufficient condition for the existence of such $F , M$, provided each $\sigma(x)$ satisfies a condition that we call ``Whitney $\omega$-convexity''.